## Geometry & Topology

### Quasigeodesic flows and sphere-filling curves

Steven Frankel

#### Abstract

Given a closed hyperbolic $3$–manifold $M$ with a quasigeodesic flow, we construct a $π1$–equivariant sphere-filling curve in the boundary of hyperbolic space. Specifically, we show that any complete transversal $P$ to the lifted flow on $ℍ3$ has a natural compactification to a closed disc that inherits a $π1$–action. The embedding $P↪ℍ3$ extends continuously to the compactification, and restricts to a surjective $π1$–equivariant map $∂P → ∂ℍ3$ on the boundary. This generalizes the Cannon–Thurston theorem, which produces such group-invariant space-filling curves for fibered hyperbolic $3$–manifolds.

#### Article information

Source
Geom. Topol., Volume 19, Number 3 (2015), 1249-1262.

Dates
Revised: 25 February 2014
Accepted: 26 July 2014
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510858761

Digital Object Identifier
doi:10.2140/gt.2015.19.1249

Mathematical Reviews number (MathSciNet)
MR3352235

Zentralblatt MATH identifier
1327.57021

#### Citation

Frankel, Steven. Quasigeodesic flows and sphere-filling curves. Geom. Topol. 19 (2015), no. 3, 1249--1262. doi:10.2140/gt.2015.19.1249. https://projecteuclid.org/euclid.gt/1510858761

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